Generic pipe dreams, lower-upper varieties, and Schwartz-MacPherson classes

2024/11/17 by Allen Knutson, Knutson, Allen, Paul Zinn-Justin +1 · 2 citations
Mathematics · #Rings, Modules, and Algebras #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications

paper · pdf · doi:10.48550/arxiv.2411.11208

Abstract

We recall the lower-upper varieties from [Knutson '05] and give a formula for their equivariant cohomology classes, as a sum over generic pipe dreams. We recover as limits the classic and bumpless pipe dream formulae for double Schubert polynomials. As a byproduct, we obtain a formula for the degree of the nth commuting variety as a sum of powers of 2. Generic pipe dreams also appear in the Segre-Schwarz-MacPherson analogue of the AJS/Billey formula, and when computing the Chern-Schwarz-MacPherson class of the orbit B_- w B₊ ⊆ Mat_{k× n} or of a double Bruhat cell B_-u B₊ ∩ B₊ v B_-.

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• Three formulas for CSM classes of open quiver loci
• Combinatorial Aspects of Elliptic Schubert Calculus

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